Project #67156 - Analysis

QUESTION 1

  1. In order to ensure the quality of its products, Umbrella Corporation regularly inspects shipments of items from its chemical supplier. In a particular box of 20 bottles of Chemical T, 3 bottles are randomly selected, without replacement, from the box. Unknown to the inspector, there are actually 4 bottles in the box that are contaminated. 

    Let G1, G2, G16 represent the “good” bottles in the box, and D1, D2, D3, and D4 represent the “bad” (contaminated) bottles. Let the random experiment be drawing 3 bottles randomly from the box, one after the other, without replacement. 

    A). Describe what the sample space S would look like, either by listing a few outcomes or describing how the outcomes would be represented. Do not attempt to list the full sample space, as there are over 1000 outcomes. 


B). what is the probability of drawing three bad items? Hint: you can use a probability tree or the multiplication rule of dependent events. You do not need to include the tree in your answer.
 



C). what is the probability of drawing two good items and one bad item? Hint: I suggest using a probability tree to make sure you find all the ways to get two good items. You do not need to include the tree in your answer.

D). Umbrella has three suppliers, A, B, and C, which supply 48%, 33%, and 19%, respectively, of its stock of another chemical, T. Supplier A has a defect rate of 3%, meaning that the probability that a batch of the chemical is defective is 0.03. The defect rates for Supplier B and C are 5% and 4%, respectively. 

The quality control inspector has tested a batch of chemical and found it to be defective. Which supplier is most likely the source of the chemical? Answer by finding the probabilities P(D|A), P(D|B), and P(D|C), P(A|D), P(B|D), and P(C|D) and choosing the supplier with the highest probability.

 

 

 

QUESTION 2

1.       Umbrella’s scientists have developed an additive that can make soft drinks sweetened with aspartame or sucralose (two popular low-calorie sweeteners) taste more like they contain real sugar, but at a fraction of the price. 

In a blind taste test, Umbrella asked 30 volunteers, independently of one another, to guess which of 3 unmarked cans contained the new sweetener. Let the random variable X be the number of successful guesses out of n = 30. In total, x = 18 out of the 30 volunteers correctly guessed the can with the new sweetener. 

a). Under the assumption that each of the 30 volunteers was randomly guessing among the three unmarked cans as to which contained the new sweetener, what is the probability that 18 or more volunteers would make a correct guess? Show all calculations. If you use Excel or other software (which I recommend), you may copy the code you use.

b). Under the assumptions of the problem, what would be the expected number of correct guesses, E(X)?


c). Use your answers in (a) and/or (b) to explain what the scientists can reasonably conclude about the effectiveness of the new sweetener.

 

 

QUESTION 3

1.       Monthly demand of units of a particular product, q, at Umbrella Corporation is random with the following distribution:

q           P(Q = q)    
4,000    0.25    
6,000    0.35    
8,000    0.40    

The price per unit is $17. Total monthly costs, T, include $7500 of fixed expenses and a per-unit cost of $8. Thus,  T = 7500 + 8Q. 
a).  Find the expected monthly demand, E(Q). Show your calculations.
b).  Let Y be the monthly profit. Find E(Y). Hints: Revenue = (price per unit)*(units sold). Profit = Revenue - Cost
c). Which approach to assigning probabilities (classical or relative frequency) would have been used to assign these probabilities?

 

 

QUESTION 4

1.       Umbrella is developing a new reagent  (a substance used to enable a specific chemical reaction) for use in testing for a certain disease. After recruiting several hundred volunteers who had previously been diagnosed as either having or not having the disease and administering the test that uses the new reagent,  the following table of estimated probabilities was constructed from the data:


                                                      Test is Positive                           Test is Negative                      Has Disease                                      0.10                                             0.02                                      Does not have disease                    0.28                                              0.60
 

 

a).   a.)The term “sensitivity” is used among scientists to refer to the “true positive rate,” that is, the probability of a positive result, given a person has the disease.  Calculate the sensitivity of the test.
b). The term “specificity” is used among scientists to refer to the “true negative rate,” that is, the probability of a negative result, given a person does not have the disease. Calculate the specificity of the test.
c). Suppose a person tests positive for the disease. What is the probability that he or she actually has the disease?

 

 

QUESTION 5

1.       A human resources (HR) director wants to investigate the nature of electronic harassment (sexual or non-sexual) in the modern workplace. She will use data from the 2012 General Social Survey (GSS) and its definition of “electronic harassment” as “any communication through emails, text messages, mobile cell phone calls, or other electronic internet or social media communications from people at work that is harassing or threatening.”

The GSS data are collected from U. S. adults 18 and over. In the data set, the frequency of harassment and the number of people who reported having been electronically harassed at that frequency over the past 12 months are shown in the table below.

Have Been

Harassed Electronically Work

Often

Sometimes

Rarely

Never

Total

FEMALE

1

11

11

541

564

MALE

1

26

6

538

571

2.      

a). Let A be the event “have been harassed either often, sometimes, or rarely at work.” Find P(A). Hint: you’ll be combining some cell counts.

b). Let B be the event “a person is FEMALE”.  Find P(B). 

c). Use either of the two definitions of independence of events to determine if it is reasonable to conclude that having been harassed is independent of gender.

 

 

 

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Due By (Pacific Time) 04/20/2015 04:00 pm
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